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These are Boolean algebra expressions. Let's break them down and simplify.

### Expression (i):
\[
F = d \cdot \overline{c} \cdot b \cdot a + d \cdot c \cdot (b + a)
\]
We can simplify this step by step.

1. Distribute \( d \cdot c \) across \( (b + a) \):
   \[
   F = d \cdot \overline{c} \cdot b \cdot a + d \cdot c \cdot b + d \cdot c \cdot a
   \]

2. Analyze for any common terms or possible simplifications. Here, there's no direct common factor in all terms. This is the simplified form.

### Expression (ii):
\[
F = c \cdot b \cdot a \cdot (c + b + d)
\]
We can simplify this as well.

1. Distribute \( c \cdot b \cdot a \) across \( (c + b + d) \):
   \[
   F = c \cdot b \cdot a \cdot c + c \cdot b \cdot a \cdot b + c \cdot b \cdot a \cdot d
   \]

2. Analyze each term:
   - The first term \( c \cdot b \cdot a \cdot c = c \cdot b \cdot a \).
   - The second term \( c \cdot b \cdot a \cdot b = c \cdot b \cdot a \).
   - The third term is \( c \cdot b \cdot a \cdot d \).

Thus, the simplified form is:
\[
F = c \cdot b \cdot a + c \cdot b \cdot a \cdot d
\]
Since \( c \cdot b \cdot a \) is common in both terms, the final simplified form is:
\[
F = c \cdot b \cdot a
\]

### Summary:
- Expression (i) simplified: \( F = d \cdot \overline{c} \cdot b \cdot a + d \cdot c \cdot b + d \cdot c \cdot a \).
- Expression (ii) simplified: \( F = c \cdot b \cdot a \).

Would you like further details or have any questions?

Here are five related questions:
1. What are the basic laws of Boolean algebra?
2. How does the distributive property work in Boolean algebra?
3. Can we simplify expressions further using Karnaugh maps?
4. What role do De Morgan’s laws play in Boolean simplification?
5. How can Boolean algebra be applied in digital logic design?

**Tip:** Always check for common factors when simplifying Boolean expressions to avoid missing simplifications.

Learn how to simplify Boolean algebra expressions such as F = d⋅c̅⋅b⋅a + d⋅c(b + a) and F = c⋅b⋅a(c + b + d) using distributive properties and Boolean algebra laws. Suitable for advanced high school and college-level students.

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